1. science

Bisimulation and public announcements in
logics of evidence-based knowledge
Bryan Renne
Computer Science, CUNY Graduate Center
[email protected]
Abstract. This paper introduces a notion of bisimulation for Artemov’s logics of evidence-based knowledge.
Bisimulation allows us to study the effect of dynamic epistemic operations on language expressivity. It is
shown that public announcements, a basic dynamic epistemic operation, add expressivity to the language
of evidenced-based knowledge. It is also shown that public announcements are definable in the language of
evidence-based knowledge augmented with an evidence admissibility relation.
1
Introduction
Plato defined knowledge as justified true belief. Following the ideas in (Hintikka 1962),
modal logics have been used as a formal means of modeling the informal notion of knowledge. If the modal is K and ϕ is a formula, then the formula Kϕ is accordingly read, “ϕ
is known.” While theories in this language can make various knowledge assertions such
as Kϕ ⊃ Kψ, the language has no means of expressing a reason as to why one assertion
follows from another, contrary to the first component of Plato’s three-part definition. Kϕ
is thus an assertion of implicit knowledge because ϕ is known for some unspecified reason.
Explicit modal logics extend the language of classical propositional logic by introducing formula-labeling terms t, allowing formation of the formula t : ϕ. In these systems, the
structure of t in a theorem t : ϕ corresponds to a particular derivation of this theorem, so
modeling knowledge using explicit modal logics naturally incorporates a notion of justification. We may thus assign to t : ϕ the reading “ϕ is known for reason t.” These logics—called
justification logics—may thus be viewed as logics of evidence-based knowlege.
So far, justification logics have only been studied in a static setting. In this paper,
we study the effect on language expressivity of public announcements, a basic dynamic
epistemic operation. Defining a notion of bisimulation appropriate for justification logics,
we show that public announcements add expressivity to the language of evidence-based
knowledge. We also show that the addition of an evidence admissibility relation to the
basic language of evidence-based knowlege makes public announcements definable within
the extended language.
1
Proceedings of the Workshop on Rationality and Knowledge, ESSLLI 2006
Sergei Artemov & Rohit Parikh (editors)
c 2006, the author(s)
Copyright 2
Justification logics
2.1
LP: Artemov’s basic logic of evidence-based knowledge
LP, Artemov’s Logic of Proofs (Artemov 2001), is the basic logic of evidence-based knowledge. The language of LP extends that of propositional logic by introducing a countable
collection of variables x1 , x2 , x3 , . . . , a countable collection of constants c1 , c2 , c3 , . . . , the
colon for forming assertions of evidence-based knowledge, the binary function symbols +
and ·, and the unary function symbol !. Terms are built up from variables and constants
using the function symbols. The rules of LP formula formation are those of propositional
logic in addition to the following: if t is a term and ϕ is an LP formula, then t : ϕ is
also an LP formula. The intended reading of t : ϕ is “t is sufficient evidence for ϕ.” The
Hilbert-style theory of LP consists of the following axiom and rule schemas:
• Classical propositional logic
C. A finite collection of axiom schemas for classical propositional logic
RC. Modus ponens: infer ψ from ϕ ⊃ ψ and ϕ
• Evidence management
LP1. t : (ϕ ⊃ ψ) ⊃ (s : ϕ ⊃ (t · s) : ψ)
LP2. t : ϕ ⊃ !t : (t : ϕ)
LP3. t : ϕ ∨ s : ϕ ⊃ (t + s) : ϕ
LP4. t : ϕ ⊃ ϕ
RLP. Constant necessitation: infer c : A from LP axiom A and constant c
For present purposes, we use Fitting’s Kripke-style semantics of (Fitting 2005), which
is based on Mkrtychev’s minimal semantics of (Mkrtychev 1997). Specifically, let an S4
Kripke model (G, Re , V ) be given.1 A function E that assigns to each world Γ and term t a
set E(Γ, t) of LP formulas is called an evidence function if it satisfies each of the following
properties:
• Evidence Closure
– Application. If ϕ ⊃ ψ ∈ E(Γ, t) and ϕ ∈ E(Γ, s), then ψ ∈ E(Γ, t · s).
– Proof Checker. If ϕ ∈ E(Γ, t), then t : ϕ ∈ E(Γ, !t).
– Sum. E(Γ, t) ∪ E(Γ, s) ⊆ E(Γ, t + s).
– Constant Specification. A ∈ E(Γ, c) for each LP axiom A and constant c.
1
G is a nonempty set of elements that are referred to as worlds, Re is a reflexive and transitive binary
relation on G, and V assigns to each world Γ a set V (Γ) of propositional letters that are the propositional
letters taken to be true at Γ.
2
• Evidence Monotonicity. If ϕ ∈ E(Γ, t) and ΓRe ∆, then ϕ ∈ E(∆, t).
Informally, E(Γ, t) is understood as the set of formulas for which t is admissible as evidence
at world Γ.2 A Fitting model is then a tuple M = (G, Re , E, V ), where E is an evidence
function. For a world Γ of a model M , we will write M, Γ |= ϕ to mean that the formula
ϕ is true at Γ in M . The negation will be written M, Γ 6|= ϕ. Truth at a world is defined
by induction on the construction of ϕ, where the propositional cases are given as usual for
Kripke models. For the LP case, M, Γ |= t : ϕ holds exactly when we have that ϕ ∈ E(Γ, t)
and that M, ∆ |= ϕ whenever ΓRe ∆.
2.2
Adding implicit knowledge
To incorporate implicit knowledge in the language of evidence-based knowledge, we wish
to extend the language of LP by introducing modals Ki for each i = 1, 2, . . . , n. We
call this extended language the language of evidence-based knowledge or, more briefly, the
EBK language. Fitting models for the EBK language are obtained from the Fitting models
defined above by adding a reflexive relation Ri corresponding to each modal Ki . Thus a
general Fitting model is a tuple M = (G, {Ri }ni=1 , Re , E, V ). In single-agent logics, where
n = 1, the subscript on both the modal and the relation will be dropped.
There are a number of ways to connect evidence-based and implicit knowledge, though
work on this has just begun. The connection studied thus far is given by the axiom schema
t : ϕ ⊃ Ki ϕ for i = 1, 2, . . . , n (Artemov 2004; Artemov and Nogina 2005). This connection
schema may be read, “An agent knows those things that have a reason.” The class of
Fitting models satisfying this connection principle is the class of Fitting models that have
Ri ⊆ Re for each Ri .
Restricting further the relation Ri , we may obtain models in which the modal Ki
behaves as in any of the epistemic logics T, S4, or S5. A justification logic corresponding
to these models may then be defined by stipulating the LP axiom and rule schemas, the
connection schema, and the corresponding modal logic axiom and rule schemas for the
modal Ki . We thus have a family of justification logics—each satisfying the connection
schema—named according to how the modality Ki behaves.3 For example, S4n LP is the
system in which each of n agents has an S4 modality. Similarly, we have Tn LP, S5n LP, and
various mixed logics in which agents’ reasoning powers differ (for example, S4S5LP is the
two-agent logic in which K1 is an S4 modal and K2 is an S5 modal).
2
To say that t is admissible (as evidence) for ϕ means that t is possible evidence for ϕ. Possible evidence
is not the same as actual evidence. If t is possible evidence for ϕ, then t may be taken into account when
considering the truth of ϕ. However, it need not be the case that t is itself sufficient to guarantee the truth
of ϕ, something we require of actual evidence. Thus the viewpoint of this paper is that (actual) evidence is
a rather strong notion because it is conclusive. This is not to say that weaker notions of evidence are not of
interest—it’s just that weaker notions have yet to be addressed because LP originated from proof-theoretic
considerations (and a proof is quite a strong notion of evidence).
3
See (Artemov 2004; Artemov and Nogina 2005; Fitting 2004) for detailed studies of this wide-ranging
family of logics.
3
3
Bisimulation for justification logics
We now define a notion of bisimulation for the EBK language. This allows us to study
expressivity issues related to dynamic epistemic operations, something we take up in the
next section of the paper.
Definition 3.1. Given the model M = (G, {Ri }ni=1 , Re , E, V ), a world Γ ∈ G, and a
formula ϕ in the EBK language, to say that ϕ is knowable at Γ means that M, ∆ |= ϕ
whenever ΓRe ∆.
Definition 3.2. Given models
M1 = (G1 , {Ri }ni=1 , Re , E1 , V1 ) and M2 = (G2 , {Si }ni=1 , Se , E2 , V2 ),
a nonempty binary relation B ⊆ P(G1 × G2 ) is a bisimulation between M1 and M2 if each
of the following conditions hold.
• The frame bisimulation conditions:
For each relation R of M1 and S of M2 both sharing the same subscript:
1. If Γ1 R∆1 and Γ1 BΓ2 , then there is a ∆2 ∈ G2 such that Γ2 S∆2 and ∆1 B∆2 ;
2. If Γ2 S∆2 and Γ1 BΓ2 , then there is a ∆1 ∈ G1 such that Γ1 R∆1 and ∆1 B∆2 .
• Agreement of propositional valuation: if Γ1 BΓ2 , then V1 (Γ1 ) = V2 (Γ2 ).
• Agreement of evidence for knowable formulas: if Γ1 BΓ2 and ϕ is knowable at Γ1 or
at Γ2 , then ϕ ∈ E1 (Γ1 , t) iff ϕ ∈ E2 (Γ2 , t) for each term t.
Two models are said to be bisimilar if there exists a bisimulation between them. World
Γ of model M and world ∆ of model N are said to be bisimilar if there is a bisimulation
B between M and N satisfying ΓB∆. For such a Γ of M and a ∆ of N , we write
(M, Γ) 'B (N, ∆), though the subscript B may be omitted when doing so ought not cause
confusion.
Remark 3.3. By restricting to the case n = 0 in Definition 3.2, we obtain bisimulation
for formulas in the the language of LP itself.
What’s new about Definition 3.2 is the condition for evidence agreement on knowable
formulas. While we could have defined bisimulation so that there is evidence agreement
for all formulas, this turns out to be too strong of a requirement because it obscures
the expressivity results we are able to obtain with this weaker notion of bisimulation.
Regardless, our notion of bisimulation is correct, as the following proposition shows.
Proposition 3.4. Let M1 and M2 be as in Definition 3.2. If (M1 , Γ1 ) 'B (M2 , Γ2 ) and ϕ
is any formula in the EBK language, then M1 , Γ1 |= ϕ iff M2 , Γ2 |= ϕ.
4
Proof. By induction on the construction of ϕ. All cases are routine except the LP inductive
case. We check this remaining case in detail, as follows. M1 , Γ1 |= t : ϕ means ϕ is knowable
at Γ1 and ϕ ∈ E1 (Γ1 , t). By the induction hypothesis, Γ1 BΓ2 implies ϕ is knowable at
Γ2 and, by the definition of bisimulation, we have ϕ ∈ E2 (Γ2 , t). Hence M2 , Γ2 |= t : ϕ.
Interchanging the models M1 and M2 in this argument gives the converse.
We now give two examples of bisimilar models, both of which will be important for
later results in the paper. The reader may wish to skip over the examples until they are
later referenced.
Example 3.5. Let p be a propositional letter, x be a variable, G = {Γ, ∆}, Re be the
smallest reflexive relation satisfying ΓRe ∆, V (Γ) = {p}, and V (∆) = ∅. We will define
models
M1 = (G, Re , E1 , V ) and M2 = (G, Re , E2 , V )
so that M1 and M2 are bisimilar, p ∈ E1 (Γ, x), and p ∈
/ E2 (Γ, x). We first specify E1 and E2
and then show that M1 and M2 are bisimilar.
Let E1 be the (unique) evidence function with the smallest graph that also satisfies
p ∈ E1 (Γ, x). It then follows that ϕ ∈
/ E1 (Γ, x) for any formula ϕ 6= p. We also have that
E1 (∆, t) = E1 (Γ, t) for all terms t.
For w ∈ G and t any term, E2 is defined as follows:
E1 (w, t) if t 6= x,
E2 (w, t) =
∅
if t = x.
It is not difficult to show that E2 is also an evidence function; most of the evidence function
properties follow immediately or else from the fact that E1 is itself an evidence function.
It’s also clear that p ∈
/ E2 (Γ, x).
M1 and M2 are clearly frame bisimulations and agree on their propositional valuations.
What remains is to show that they also satisfy the condition on the evidence functions
E1 and E2 . So suppose that ϕ is knowable at Γ in M1 . Certainly it cannot be the case
that ϕ is p, for p is not knowable at Γ in M1 . Thus we have ϕ ∈
/ E1 (Γ, x) because ϕ 6= p,
and we also have ϕ ∈
/ E2 (Γ, x) because E2 (Γ, x) = ∅. So, in the case t = x, the evidence
functions agree on the knowable formula ϕ. In case t 6= x, then we have E1 (Γ, t) = E2 (Γ, t)
by definition, so the evidence functions clearly agree on the knowable formula ϕ. The case
where ϕ is knowable at Γ in M2 is shown in the same way. A similar argument also applies
at the world ∆. Hence M1 and M2 are bisimilar.
Example 3.6. Let G = {Γ}, Re = {(Γ, Γ)}, and V (Γ) = ∅. Let x be a variable. Then
there are models
M1 = (G, Re , E1 , V ) and M2 = (G, Re , E2 , V )
such that M1 and M2 are bisimilar, ⊥ ∈ E1 (Γ, x), and ⊥ ∈
/ E2 (Γ, x). The construction
of the evidence functions E1 and E2 is analogous to that given in Example 3.5, as is the
verification of the bisimulation condition on the evidence functions.
5
4
Public announcements and expressivity
A public announcement of the formula ϕ operates on an epistemic model by deleting all
those worlds in which ϕ does not hold (Plaza 1989). Public announcements appear as
labeled modalities: [ϕ]ψ means that ψ holds after the public announcement of ϕ. If L is
an epistemic language, then the language L with public announcements is the extension
of L obtained by adding brackets (for formation of public announcement formulas) and
admitting an additional rule of formula formation for public announcement formulas: if ϕ
and ψ are formulas, then so is [ϕ]ψ. We now define the truth of a public announcement
formula at a world of a model.
Definition 4.1. If Γ is a world of the model M = (G, {Ri }ni=1 , Re , E, V ), then M, Γ |= [ϕ]ψ
means that either M, Γ 6|= ϕ or that M |ϕ, Γ |= ψ, where M |ϕ is the submodel of M obtained
by deleting all those worlds of M in which ϕ does not hold. That is,
M |ϕ := (Gϕ , {Riϕ }ni=1 , Reϕ , E ϕ , V ϕ )
where
• Gϕ := {Γ ∈ G | M, Γ |= ϕ}
• Riϕ := Ri ∩ (Gϕ × Gϕ ) for i = 1, 2, . . . , n
• Reϕ := Re ∩ (Gϕ × Gϕ )
• E ϕ (∆, t) := E(∆, t) for ∆ ∈ Gϕ and t a term
• V ϕ (Γ) := V (Γ) for Γ ∈ Gϕ
This definition also works for the public announcement of ϕ in a multi-agent Kripke model
M —a model that does not contain the evidence function E—by omitting the mention of
evidence functions.
Lemma 4.2 (Correctness). Let ϕ be a formula in the EBK language with public announcements. If Γ is a world of a Fitting model M and M, Γ |= ϕ, then M |ϕ is a Fitting
model.
Proof. A straightforward verification.
Suppose that T is an epistemic theory that is sound and complete with respect to a
fixed class of Kripke models. To say that public announcements are definable within T
means that for every formula ϕ in the language of T with public announcements, there is a
formula ψ in the language of T without public announcements such that the biconditional
6
ϕ ≡ ψ is valid. Plaza was the first to show that public announcements are definable within
S5n (Plaza 1989), and this result extends naturally to the epistemic logics Tn and S4n .4
The next theorem exhibits a formula in the language of LP with public announcements
that is equivalent to no formula in the language of LP without public announcements.
Therefore, public announcements are not definable within LP.
Theorem 4.3. The language of LP with public announcements is strictly more expressive
than the language of LP.
Proof. In the models M1 and M2 of Example 3.5, we have that (M1 , Γ1 ) ' (M2 , Γ2 ), from
which it follows by Proposition 3.4 that no LP formula distinguishes Γ1 and Γ2 . Since
[p]x : p holds at Γ1 and not at Γ2 , we have that [p]x : p is equivalent to no LP formula.
Since our justification logics extend LP, this theorem extends naturally to justification
logics in the EBK language. Therefore, public announcements are not definable in any
justification logic. This is the statement of following corollary.
Corollary 4.4. The EBK language with public announcements is strictly more expressive
than the EBK language (without public announcements).
Proof. Let M10 be the trivial extension of the model M1 of Theorem 4.3; that is, Ri =
{(Γ, Γ)} for i = 1, 2, . . . , n. Define M20 similarly. The same phenomenon then occurs with
the formula [p]x : p.
As Evan Goris observed, the formula [p]x : p has an interesting interpretation that is
summarized by the following proposition.
Proposition 4.5. Let Γ be a world of a model M = (G, Re , E, V ). Then
M, Γ |= [p]x : p iff M, Γ |= p implies p ∈ E(Γ, x) .
Proof. Suppose that both M, Γ |= [p]x : p and M, Γ |= p. It then follows from the definition
of truth of a public announcement formula at a world that M |p, Γ |= x : p. This implies
p ∈ E p (Γ, x) by the definition of truth of a formula of the form x : ϕ. But then p ∈ E(Γ, x)
because E p (Γ, x) = E(Γ, x) by definition.
4
The result breaks down in multi-agent logics with common knowledge (Baltag, Moss, and Solecki
1999; Baltag, Moss, and Solecki 2005), though Kooi and van Benthem show in (Kooi and van Benthem
2004) that public announcements are again definable if the language of epistemic logic with common
knowledge is extended by introducing a notion of relativized common knowledge (see also their paper (van
Benthem, van Eijck, and Kooi 2005) with van Eijck). Whereas the (unary) common knowledge modality C
behaves as reachability—M,
Γ |= Cϕ iffSϕ holds in each world reachable from Γ via a path in the reflexive
Sn
∗
n
transitive closure of i=1 Ri , written ( i=1 Ri ) —relativized common knowledge is a binary modality C r
that behaves as
reachability—C r (ϕ, ψ) holds iff ψ holds in each world reachable from Γ via a
Snrestricted
∗
path path in ( i=1 Ri ) whose worlds all satisfy ϕ. Then Cϕ ≡ C r (>, ϕ) is valid, and it can be shown
that public announcements are definable in the extended language containing C r .
7
Conversely, suppose M, Γ |= p implies p ∈ E(Γ, x). In the case M, Γ 6|= p, we have
M, Γ |= [p]x : p trivially, so assume M, Γ |= p and thus that p ∈ E(Γ, x). To see that
M |p, Γ |= x : p, it remains to be shown that p is knowable at Γ in M |p. But this follows
immediately from the fact that p holds at every world of M |p by definition. The result
follows.
So the language of LP with public announcements can describe evidence admissibility
for some true formulas, though evidence admissibility is usually a strictly semantic notion.
This perhaps provides the reader with some intuition as to why public announcements add
expressivity to the EBK language.
5
Adding assertions of evidence admissibility
We have seen that adding public announcements to the EBK language adds expressivity,
and hence public announcements are not definable within any of our justification logics.
In the present section, we address this matter by defining the theory EJ , a conservative
extension of LP in which public announcements are definable. We will then describe how
nautral extensions of EJ do the job for arbitrary justification logics.
5.1
EJ : a basic logic with public announcement definability
It is now our task to provide a conservative extension of LP in which public announcements
are definable. Proposition 4.5 suggests that it might be sufficient to expand the language
to include explict assertions of evidence admissibility. This is the route we shall take.
If t is a term and ϕ is a formula, then we introduce the new formula t ϕ whose
intended reading is “t is admissible (as evidence) for ϕ.” t ϕ is true in world Γ of model
M = (G, Re , E, V ) exactly when ϕ ∈ E(Γ, t).
For purposes of technical simplicity, we will also add an S4 modal J to the language.
This makes it straightforward both to express t : ϕ in terms of evidence admissibility and
also to capture Evidence Monotonicity.
All together, the language of EJ is obtained from that of LP by adding the symbols and J. The EJ rules of formula formation are those of LP in addition to the following: if
t is a term and ϕ is an EJ formula, then both t ϕ and Jϕ are also EJ formulas. The
Hilbert-style theory of EJ consists of the following axiom and rule schemas:
• Classical propositional logic
A. Finite number of axiom schemas for classical propositional logic
RA. Modus ponens: infer ψ from ϕ and ϕ ⊃ ψ
• S4 knowledge for J
J1. J(ϕ ⊃ ψ) ⊃ (Jϕ ⊃ Jψ)
8
J2. Jϕ ⊃ ϕ
J3. Jϕ ⊃ JJϕ
RJ. J necessitation: infer Jϕ from ϕ
• Evidence admissibility
E1. (t (ϕ ⊃ ψ)) ⊃ ((s ϕ) ⊃ ((t · s) ψ))
E2. (t ϕ) ⊃ (!t t : ϕ)
E3. (t ϕ) ∨ (s ϕ) ⊃ ((t + s) ϕ)
E4. (t ϕ) ⊃ J(t ϕ)
RE. Infer c A from axiom A and constant c
• Connection principle
C. t : ϕ ≡ Jϕ ∧ (t ϕ)
Proposition 5.1. EJ is a conservative extension of LP and of S4.
Proof. EJ is clearly sound for LP Fitting models. EJ is also complete for these models,
as we show in a moment via a canonical model construction. This then gives the desired
result. So we proceed with the construction. Note that by sets we mean sets of EJ formulas.
A set is consistent if for no finite subset is ⊥ provable. Any consistent set may be
extended to a maximal consistent set as usual. For convenience, if Γ is a set, let Γ# :=
{ϕ | Jϕ ∈ Γ}. Now define the canonical model M = (G, Re , E, V ) in the usual way: G
is the set of all maximal consistent sets, we have ΓRe ∆ if and only if Γ# ⊆ ∆, we set
E(Γ, t) := {ϕ | t ϕ ∈ Γ}, and we set V (Γ) := {p | p ∈ Γ}.
To verify that M is a Fitting model, two items must be checked. That (G, Re , V ) is
an S4 Kripke model is straightforward. That E is an evidence function follows from the
definitions of E and Re in M and the axiom schemas E1, E2, E3, and E4.
We then verify a property of M called the Truth Lemma: ϕ ∈ Γ if and only if M, Γ |= ϕ.
This then immediately yields completeness: if ϕ is not provable, then {¬ϕ} is consistent
and may be thus be extended to a world Γ of M . Applying the Truth Lemma, we have
that M, Γ 6|= ϕ, as desired. So what remains is the proof of the Truth Lemma.
The proof is by induction on the construction of ϕ. Most cases are standard, following
from the definition of M and the induction hypothesis. We will handle the case t : ϕ and
leave the rest for the reader.
If t : ϕ ∈ Γ, then Jϕ ∈ Γ by C, and thus ΓRe ∆ implies ϕ ∈ ∆ and hence M, ∆ |= ϕ by
the induction hypothesis. Since t : ϕ ∈ Γ also implies t ϕ ∈ Γ by C, we have ϕ ∈ E(Γ, t).
Thus M, Γ |= t : ϕ.
If ¬t : ϕ ∈ Γ, then ¬Jϕ ∈ Γ or ¬(t ϕ) ∈ Γ by C. In the latter case, ϕ ∈
/ E(Γ, t), so
M, Γ 6|= t : ϕ, as desired. In the case ¬Jϕ ∈ Γ, we claim that Γ# ∪{¬ϕ} is consistent. Were it
not, then for a finite Γ1 ⊆ Γ# , we would have EJ ` !Γ1 ⊃ ϕ, and thus EJ ` J(!Γ1 ) ⊃ Jϕ.
Since J is an S4 modal and Γ1 ⊆ Γ# , it follows that J(!Γ1 ) ∈ Γ, and thus Jϕ ∈ Γ,
9
contradicting the consistency of Γ. Hence Γ# ∪{¬ϕ} is consistent and thus may be extended
to a world ∆ of M . We then have M, ∆ 6|= ϕ by the induction hypothesis, and, since ΓRe ∆,
we have shown M, ∆ 6|= t : ϕ, as desired.
Proposition 5.2. Public announcements are definable within EJ .
Proof. A complete list of reduction schemas for EJ is as follows.
[ϕ]p
[ϕ](ψ ⊃ χ)
[ϕ]Jψ
[ϕ][ψ]χ
[ϕ]t : ψ
[ϕ](t ψ)
≡
≡
≡
≡
≡
≡
ϕ⊃p
[ϕ]ψ ⊃ [ϕ]χ
ϕ ⊃ J[ϕ]ψ
[ϕ ∧ [ϕ]ψ]χ
ϕ ⊃ (J[ϕ]ψ ∧ (t ψ))
ϕ ⊃ (t ψ)
Here p is an atom. Each schema is valid for LP Fitting models.
We saw in Proposition 4.5 that the language of LP with public announcements can
describe evidence admissibility for some true formulas. Since EJ can do so for all formulas,
whether true or not, it seems as though EJ can say more. This is in fact the case, as the
following theorem shows.
Theorem 5.3. The language of EJ is strictly more expressive than that of LP with public
announcements.
Proof. We will prove the following fact: no formula ϕ in the language of LP with public
announcements can distinguish the models M1 and M2 of Example 3.6. Since we have that
x ⊥ holds at Γ in M1 but not at Γ in M2 , it then follows that x ⊥ is equivalent to
no formula in the language of LP with public announcements. So what remains is to prove
the above-stated fact; we do this by induction on the construction of ϕ. The base and
propositional inductive cases are straightforward, so we handle only the other two cases.
Suppose M1 , Γ |= t : ϕ. This implies M1 , Γ |= ϕ and thus M2 , Γ |= ϕ by the induction
hypothesis. It follows that ϕ 6= ⊥, and so ϕ ∈ E1 (Γ, t) if and only if ϕ ∈ E2 (Γ, t) by the
construction of E2 . But since we have that ϕ ∈ E1 (Γ, t) from our assumption M1 , Γ |= t : ϕ,
it follows that ϕ ∈ E2 (Γ, t). We have then shown that M2 , Γ |= t : ϕ, as desired.
Suppose M1 , Γ 6|= t : ϕ. Then we have M1 , Γ 6|= ϕ or ϕ ∈
/ E1 (Γ, t). If M1 , Γ 6|= ϕ, then
M2 , Γ 6|= ϕ by the induction hypothesis and thus M2 , Γ 6|= t : ϕ, as desired. If M1 , Γ |= ϕ and
ϕ∈
/ E1 (Γ, t), then ϕ 6= ⊥ and so ϕ ∈
/ E2 (Γ, t) by the construction of E2 . Thus M2 , Γ 6|= t : ϕ.
For the inductive case [ϕ]ψ, notice that [ϕ]ψ ≡ ϕ ⊃ ψ is a valid scheme in any one-world
model. This inductive case is thus handled by the propositional inductive case.
5.2
Adding implicit knowledge
To extend LP so as to incorporate implicit knowledge, we added to the language of LP a
T, S4, or S5 modal K and to the theory the schema t : ϕ ⊃ Kϕ. Similarly, we will now
10
add implicit knowledge to EJ in order to extend arbitrary justification logics to ensure that
public announcements are definable. We will show how to do this with a single modal K,
which may be T, S4, or S5. This yields the theories TEJ (extending TLP), S4EJ (extending
S4LP), and S5EJ (extending S4LP). Iterating the process to add additional modals obtains
an appropriate extension of the corresponding justification logic. Proceeding, we address
the case where we are adding an S4 modal K. The cases where K is T or S5 are handled
analogously.
The language of S4EJ is obtained from that of EJ by adding the new symbol K. The
rules of S4EJ formula formation are those of EJ in addition to the following: if ϕ is an
S4EJ formula, then Kϕ is also an S4EJ formula. The Hilbert-style theory of S4EJ consists
of the following axiom and rule schemas:
• Axiom and rule schemas for EJ
• S4 knowledge for K
• J-K connection principle
C2. Jϕ ⊃ Kϕ
Proposition 5.4. S4EJ is a conservative extension of S4LP.
Proof. S4EJ Fitting models are just S4LP Fitting models. These models have the form
M = (G, R, Re , E, V ), where R ⊆ Re . To interpret EJ formulas, R interprets K and Re
interprets both J and assertions of the form t : ϕ. Since R ⊆ Re , we see that S4EJ is clearly
sound for these models. That S4EJ is complete for these models follows by extending the
EJ canonical model construction in the obvious way to incorporate the modal K and then
verifying the additional inductive case for Kϕ in the Truth Lemma. The latter verification
makes use of the J-K connection principle (and, in particular, need not make use of the
fact that K is S4, and thus we could just as easily have chosen K to be T or S5).
So S4EJ is sound and complete for S4LP Fitting models. Thus a S4EJ theorem ϕ in
the language of S4LP is an S4LP validity and is thus also an S4LP theorem.
Proposition 5.5. Public announcements are definable within S4EJ .
Proof. Add [ϕ]Kψ ≡ ϕ ⊃ K[ϕ]ψ to the list of reduction schemas in Proposition 5.2 to
obtain a complete list of reduction schemas for S4EJ . These schemas are all valid for S4EJ
Fitting models.
Theorem 5.6. The language of S4EJ is strictly more expressive than that of S4LP with
public announcements.
Proof. As in the case for EJ (Theorem 5.3). The extra inductive case is trivial because
Kϕ ≡ ϕ in one-world models by the reflexivity of K.
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